Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #471 |
104 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,8)(7,12)(9,15)(10,16)(11,18)(13,21)(14,17)(19,20);(6,7,9,11,10,13,17,19)(8,12,15,18,16,21,14,20);(2,3)(4,5)(7,11)(8,14)(9,17)(13,19)(15,16)(18,20)Polycyclic presentation:
2^6,5|b^2=e,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^4,d^c=df56 conjugacy classes — view the full character table.
| maximal subgroup of | #10611623, #10611624, #10611631, … |
| maximal normal subgroup of | #10611623, #10611624, #10611631, … |
| maximal quotient of | #10600684, #10600685, #10600686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 768);